
because \(\psi _H\)  is bounded by \(\beta \) . Then, we use Lemma REF  to bound the sum of i.i.d. bounded random variables. Having \(\psi _H\)  bounded, we see that by Hoeffding's lemma (see [1]}), \(\max _{1\le i\le n}\Vert \psi _H(\Vert X_i-T_H(P)\Vert )\Vert _{\psi _1}\le \beta \) . Then, by Lemma REF  for all \(t>0\) ,
\(\mathbb {P}\left(\left\Vert \frac{1}{n}\sum _{i=1}^n \frac{X_i^{\prime }-T(P)}{\Vert X_i^{\prime }-T_H(P)\Vert }\psi _H\left(\Vert X_i^{\prime }-T_H(P)\Vert \right) \right\Vert  \ge \frac{3}{2}\sqrt{\frac{V_H}{n}}+2\sqrt{\frac{tv_H}{n}}+\frac{Ct\beta }{n}\right)\le 4e^{-t} ,\) 
